Projective Geometry - Lecture Notes Chapter 4

We define a new geometry based on a Klein quadratic set.

Let C and C* be the two equivalence classes of planes of a Klein quadratic set Q in a 5-dimensional projective space P. We define the geometry S as follows:

1. the points of S are the planes of C.
2. the lines of S are the points of Q.
3. the planes of S are the planes of C*.
4. the incidence between a line of S and either a point or plane of S is induced by the incidence in P.
5. a point ₁ of S is incident with a plane ₂ of S iff the planes ₁ and ₂ of P are not disjoint (then by 4.5.5 they intersect each other in a line of P).

Lemma 4.6.1: Let Q be a Klein quadratic set.

1. Each Q-line is on exactly one plane of each equivalence class.
2. If P is finite of order q, then each point of Q is on exactly q+1 planes of each equivalence class.

Theorem 4.6.2: The geometry S is a 3-dimensional projective space; more precisely, S is isomorphic to a 3-dimensional subspace of P.

4.7 Quadrics

1. q(av) = a²q(v) for all v in V and a in F.
2. the map B: V × V -> F defined by B(v,w) = q(v+w) - q(v) - q(w) is a symmetric bilinear form.

Examples:

Consider the 3 dimensional vector space V over the reals.

Consider the 3 dimensional vector space V over the reals.

Lemma 4.7.1: Let {v₁,...,v_n} be a basis of the vector space V.

1. If a_ij in F, then a quadratic form is defined by the following rule: q( b_i v_i ) = a_ij b_i b_j.
2. Conversely, for any quadratic form q there are elements a_ij in F such that for all v in V we have the above form.

Def: A quadratic form is nondegenerate if q(v) = 0 and B(x,v) = 0 for all x in V implies that v = O.

Def: Let q be a quadratic form of the vector space V. The quadric of the projective space P(V) corresponding to q is the set of all points <v> of P(V) with q(v) = 0.

Examples:

By Theorem 2.4.4 the points on a regulus in a 3-dimensional projective space can be given coordinates that satisfy,

The quadratic form x₀² – x₁² - x₂² yields a quadric in the plane which is an oval. The quadratic form x₀² – x₁² + x₂² yields a quadric in the plane which is the union of two lines.

In a 3-dimensional projective space, the quadric given by x₀² - x₁² – x₂² – x₃² is an ovoid, while the quadric given by x₀² + x₁² - x₂² - x₃² is a hyperboloid.

Lemma 4.7.2: Let q be a quadratic form of a vector space V, and let Q be the corresponding quadric in P(V). Then, if a line g contains three points of Q, each point of g lies in Q.

Def: Let q be a quadratic form of the vector space V. For a non-zero vector v in V we define

Lemma 4.7.3: Let q be a quadratic form of the vector space V, and let Q be the corresponding quadric of P(V). Then for each non-zero vector v in V we have:

1. <v>^perp is a subspace of V, hence also a subspace of P(V).
2. <v>^perp is either a hyperplane or equal to V.
3. Suppose that <v> in Q. Then each line of <v>^perp through <v> is a tangent line of Q.
4. Suppose that <v> in Q. Then each line through <v> that is not contained in <v>^perp intersects Q in precisely one further point.

Theorem 4.7.4: Each quadric is a quadratic set.

A fundamental theorem of Buekenhout states that any nondegenerate quadratic set is either a quadric or an ovoid. We have already proved this for hyperbolic sets in 3-dimensional projective spaces (by showing that these non-ovoids are quadrics.) In the next section we prove an analogous statement for Klein quadratic sets.

In the case d = 2 the corresponding theorem was first proved by B. Segre. In the case d = 3 this was first done by Barlotti and Panella. Segre's theorem is particularly remarkable.

Theorem 4.7.5: Any oval in a finite Desgarguesian projective plane of odd order is a conic (a nonempty, nondegenerate quadric in a projective plane).